We denote $B$ the function defined on $\mathbb { R } ^ { + * }$ by $B ( x ) = \frac { \partial ^ { 2 } \beta } { \partial y ^ { 2 } } ( x , 1 )$, where $\beta ( x , y ) = \int _ { 0 } ^ { 1 } t ^ { x - 1 } ( 1 - t ) ^ { y - 1 } \mathrm {~d} t$ and $\psi ( x ) = \frac { \Gamma ^ { \prime } ( x ) } { \Gamma ( x ) }$. We have $\frac { \partial \beta } { \partial y } ( x , y ) = \beta ( x , y ) ( \psi ( y ) - \psi ( x + y ) )$. Justify that $B$ is defined on $\mathbb { R } ^ { + * }$. Using the relation found in III.B.1, establish that for every real $x > 0$ $$x B ( x ) = ( \psi ( 1 + x ) - \psi ( 1 ) ) ^ { 2 } + \left( \psi ^ { \prime } ( 1 ) - \psi ^ { \prime } ( 1 + x ) \right)$$ Deduce that $B$ is $\mathcal { C } ^ { \infty }$ on $\mathbb { R } ^ { + * }$.
We denote $B$ the function defined on $\mathbb { R } ^ { + * }$ by $B ( x ) = \frac { \partial ^ { 2 } \beta } { \partial y ^ { 2 } } ( x , 1 )$, where $\beta ( x , y ) = \int _ { 0 } ^ { 1 } t ^ { x - 1 } ( 1 - t ) ^ { y - 1 } \mathrm {~d} t$ and $\psi ( x ) = \frac { \Gamma ^ { \prime } ( x ) } { \Gamma ( x ) }$. We have $\frac { \partial \beta } { \partial y } ( x , y ) = \beta ( x , y ) ( \psi ( y ) - \psi ( x + y ) )$.
Justify that $B$ is defined on $\mathbb { R } ^ { + * }$.
Using the relation found in III.B.1, establish that for every real $x > 0$
$$x B ( x ) = ( \psi ( 1 + x ) - \psi ( 1 ) ) ^ { 2 } + \left( \psi ^ { \prime } ( 1 ) - \psi ^ { \prime } ( 1 + x ) \right)$$
Deduce that $B$ is $\mathcal { C } ^ { \infty }$ on $\mathbb { R } ^ { + * }$.